Group #26132660 — subgroups

Subgroups — #26132660

Order 12032.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#20243552	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal
4	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093956	—	2	1	no	—
8	#15688989	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₄₇	C₂²	1	2	yes	—
11	C₂×#106×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	#13093958	—	2	4	no	—
19	#13093956	—	4	2	no	—
20	C₄₇⋊#95	—	4	2	no	—
21	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
25	#107×C₄₇	—	2	4	no	—
26	#106×C₄₇	—	4	2	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₄₇	Q₈	1	2	yes	—
29	#95×C₄₇	—	4	2	no	—
30	C₂²×C₄×C₄₇	—	4	2	no	—
31	C₂²×C₄×C₄₇	D₈	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₄₇	Q₁₆	1	2	yes	—
45	C₄⋊Dic₄₇	—	4	2	no	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	C₄₇⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	—	8	2	no	—
51	C₂×C₄×C₄₇	—	8	4	no	—
52	C₂×C₄×C₄₇	#107	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₄₇	—	8	2	no	—
60	C₂×Dic₄₇	—	8	4	no	—
61	C₈×C₄₇	#106	1	4	yes	—
62	C₈×C₄₇	#117	1	2	yes	—
63	C₈×C₄₇	#125	1	2	yes	—
64	#12889	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
66	C₂²×C₄₇	#486	1	1	yes	characteristic
67	C₂²×C₄₇	#486	1	1	yes	characteristic
68	C₂²×C₄₇	#508	1	4	yes	—
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	—	16	2	no	—
71	C₄×C₄₇	#382	1	1	yes	characteristic
72	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
73	C₄×C₄₇	#487	1	1	yes	characteristic
74	C₄×C₄₇	#487	1	1	yes	characteristic
75	Dic₄₇	—	16	2	no	—
76	C₂×C₄×C₁₆	—	47	1	no	—
77	#1846	—	47	1	no	—
78	#1846	D₄₇	1	1	yes	characteristic
79	#1380	—	47	4	no	—
80	C₂×C₄₇	#1834	1	1	yes	characteristic
81	C₂×C₄₇	#2158	1	2	yes	—
82	C₂×C₄₇	#2242	1	2	yes	—
83	C₂×C₄₇	#2242	1	2	yes	—
84	C₂²×C₁₆	—	47	2	no	—
85	C₂×#107	—	47	2	no	—
86	C₂×#107	C₂×D₄₇	1	2	yes	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	—	94	1	no	—
89	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
90	#382	—	2	1	no	—
91	#382	—	94	1	no	—
92	C₄×C₁₆	—	47	4	no	—
93	C₄₇	#12889	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	94	2	no	—
98	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₂×C₁₆	—	47	4	no	—
100	C₂×C₁₆	—	94	2	no	—
101	C₂×C₁₆	—	94	2	no	—
102	#107	—	2	4	no	—
103	#107	—	94	4	no	—
104	#106	—	4	2	no	—
105	#106	—	188	2	no	—
106	C₄×C₈	Dic₉₄	1	2	yes	—
107	C₄×C₈	C₄×D₄₇	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	188	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	188	2	no	—
112	C₂²×C₄	#10625334	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
115	C₂×C₈	#10625335	1	2	yes	—
116	C₂×C₈	#10625336	1	2	yes	—
117	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
119	C₂×C₈	#10625352	1	2	yes	—
120	C₂×C₈	#10625354	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₄⋊C₄	—	188	2	no	—
125	C₄²	Dic₄₇⋊C₄	1	2	yes	—
126	C₄²	#10625337	1	2	yes	—
127	C₁₆	—	94	4	no	—
128	C₂³	C₂×#10625334	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	376	4	no	—
133	C₂×C₄	—	376	2	no	—
134	C₂×C₄	#13093957	1	4	yes	—
135	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#13094011	1	1	yes	characteristic
137	C₂×C₄	#13094013	1	1	yes	characteristic
138	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
139	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
140	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
141	C₈	#13093955	1	4	yes	—
142	C₈	#13093982	1	2	yes	—
143	C₈	#13094015	1	2	yes	—
144	C₂²	C₂×#13093957	1	1	yes	characteristic
145	C₂²	#15689469	1	1	yes	characteristic
146	C₂²	#15689471	1	1	yes	characteristic
147	C₂²	#15689553	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	752	2	no	—
151	C₄	#15688988	1	1	yes	characteristic
152	C₄	C₂×#13093955	1	1	yes	characteristic
153	C₄	#15689473	1	1	yes	characteristic
154	C₄	#15689475	1	1	yes	characteristic
155	C₂	#20243515	1	1	yes	characteristic
156	C₂	#20245170	1	2	yes	—
157	C₂	#20245619	1	2	yes	—
158	C₂	#20245622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

#20243552

C₂

yes

maximal, characteristic

#20241479

C₂

yes

maximal

C₄₇×#1846

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

#15688989

—

C₄×C₄₇⋊C₁₆

C₄

yes

complemented

C₂×#107×C₄₇

C₂²

yes

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#382

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093958

—

#13093956

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#117

yes

—

C₈×C₄₇

#125

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₄₇

yes

characteristic

#1380

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

#2158

yes

—

C₂×C₄₇

#2242

yes

—

C₂×C₄₇

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₄₇

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#382

—

#382

—

C₄×C₁₆

—

C₄₇

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

#107

—

103

#107

—

104

#106

—

105

#106

—

188

—

106

C₄×C₈

Dic₉₄

yes

—

107

C₄×C₈

C₄×D₄₇

yes

—

108

#95

—

109

#95

—

188

—

110

C₂²×C₄

—

111

C₂²×C₄

—

188

—

112

C₂²×C₄

#10625334

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

114

C₂×C₈

Dic₄₇⋊C₄

yes

—

115

C₂×C₈

#10625335

yes

—

116

C₂×C₈

#10625336

yes

—

117

C₂×C₈

C₂×Dic₉₄

yes

characteristic

118

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

119

C₂×C₈

#10625352

yes

—

120

C₂×C₈

#10625354

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

188

—

124

C₄⋊C₄

—

188

—

125

C₄²

Dic₄₇⋊C₄

yes

—

126

C₄²

#10625337

yes

—

127

C₁₆

—

128

C₂³

C₂×#10625334

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

376

—

133

C₂×C₄

—

376

—

134

C₂×C₄

#13093957

yes

—

135

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

136

C₂×C₄

#13094011

yes

characteristic

137

C₂×C₄

#13094013

yes

characteristic

138

C₂×C₄

C₂×#10625336

yes

characteristic

139

C₂×C₄

C₂×#10625335

yes

characteristic

140

C₂×C₄

C₂×#10625337

yes

characteristic

141

C₈

#13093955

yes

—

142

C₈

#13093982

yes

—

143

C₈

#13094015

yes

—

144

C₂²

C₂×#13093957

yes

characteristic

145

C₂²

#15689469

yes

characteristic

146

C₂²

#15689471

yes

characteristic

147

C₂²

#15689553

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

752

—

151

C₄

#15688988

yes

characteristic

152

C₄

C₂×#13093955

yes

characteristic

153

C₄

#15689473

yes

characteristic

154

C₄

#15689475

yes

characteristic

155

C₂

#20243515

yes

characteristic

156

C₂

#20245170

yes

—

157

C₂

#20245619

yes

—

158

C₂

#20245622

yes

—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1846	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#12889
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1846

C₂

socle, prime radical

C₂³×C₄₇

C₂×D₈

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₄₇⋊C₁₆	C₄	semidirect
93. C₄₇	#12889	semidirect

H ≅ G/N

type

9. C₄×C₄₇⋊C₁₆

C₄

semidirect

93. C₄₇

#12889

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
#20243552	2
#20241479	2
C₄₇×#1846	2
#12889	47

subgroup

index

C₂×C₄×C₄₇⋊C₁₆

#20243552

#20241479

C₄₇×#1846

#12889

Subgroups — #26132660

Conjugacy classes of subgroups

Complements

Maximal subgroups